【实例简介】

The paper presents an efficient 88 line MATLAB code for topology optimization. It has been developed using the 99 line code presented by Sigmund (Struct Multidisc Optim 21(2):120-127, 2001) as a starting point. The original code has been extended by a density filter, and a considerable improvement i

E

C

Ee(e)= Emin +aP(Eo-Emin), e E[0,1

dr

dV

d e

av

min c(x)=UTKU=>Ee(ae) koue

V(x/Vo=f

KU=F

U

F

dc/ x

H

H

i∈N

△(c,)

max

eB< max(O

Hei= max(0, rminA(c, i)

new

Inn

xeB≥min(1,xe

(=10-3

B

7(=1/2

∑Hx

2∈N

Ph

已

0元e

0

H

2n-1

2

top88(ne⊥x,nely,

o⊥fra

nal,xmin,王

melt

nely

f penal

p rmin

rnin

lIe

ft 2

EO

Emin

top88(60,20,0.5,3,1.5,1)

for

edofMat

sK

edofMat

Ee(e

(nely+1)×(nelx+1)

nodenrs

reshape

sparse

moderns

K

sparse

edofvec

edofvec

sparse

repmat

repmat

edofve

edor vec

format

3411129101

561314111234+

edofMat=111219201718910←

3132394037382930←

K

sparse

for

sparse

ac/a

max(,e)∑∈NHa

dc/a

H

dofMat

H

K

K(k

(,j)

k=2+8(1-1)

H

Hs

(nelx x nely)x(nelx x nely)

H

lmid

H

spars

GH

JH

sH

dy

H JH

H

H

H

iH JH

SH

150×50

300×100

50%

edofmat

edofmat

edofMat

edofMat

C=21681

C=219.5

222.29

C=233.71

C=235.73

C=238.31

60×20

150×50

300×100

∑

H(m2)(k-m,-m)

升(m,n)

conv

H(m, n)

conv

H(m, n)=max(O, Tmin -f(m, n))

H(m, n)

b-m

m, n

M×N

h(m+4+n+4+)=x(m,2)

∑h(m+,+(k-m

m.7

k, l)

+

Tn.ri

h

conv2(xh,'same))

[dy, dx] meshgrid(-ceil (rmin)+1: ceil(rmin)-1,

ceil(rmin)+1: ceil(rmin)-1)

h = max(o, rmin-sqrtdr "2+dy. 2))

Hs=conv2(ones(nely,nelx),h,’same’)

h

LILl

IiIL

Hs

Rmin

f ft

ic conv2(dc. *xPhys,h,'same,)./Hs /max(1e-3,xPhys)

elseif ft

dc conv 2(dc/

dv conv(dv/Hs, h, 'same)

xPhys conv2 (xnew, h, 'same?)./Hs;

KFXN TEX

F

x=T文

raIn

0

Rmin rmin/2/sgrt(3)

KEF=Rmin^2*[4-1-2-1;-14-1-2

2-14-1;-1-2-14]/6

[4212;242

1242;2124]/36;

edofVecF reshape(nodenrs(1: end-1, 1: end-1), nelx*nely, 1);

edofMatF repmat (edofVecF,1,4)+

min

[1:2]1],ne1x*ne1y,1)

iKF= reshape (kron(edofMatF, ones(4, 1))', 16*nelx*nely, 1);

jKF reshapekron(edofMatF, or

sKF reshape(KEF(: )*ones (1, nelx*nely), 16*nelxxnely, 1);

Rmin=rmin/2√3

KF = sparse(iKF, jKF, sKF)

LF chol(KF, 'lower)

iTF- reshape(edofMatF, 4*nelx*nely, 1)

JTF reshape(repmat([1: nelx*nely, 4, 1),4*nelx*nely, 1);

at(1/4,4*ne1x*ne1y,1)

TF sparse(iTF, jTF, sTF);

19.44

C=237.60

C=235.36

C=236.62

60×20

150×50

00×10

KF

TF

ff

dc(:)=(TF*(LF\(LF\(TF*(dc(:),*xFhys(;))),

/max(18-3, xPhys(: ))

Seif ft = 2

dc(: )=TF*(LF,(LF\(TFxdc(: )))

dv(:)=TF:*(LF\(LF\(TF*dv(:))))

d

Phys(:)=(TF]*(LF\(LF\(TF*]new(:))))

F= sparse(2*(ney+1)*(ne1x+1),1,-1,

2*(ne1y+1)*(ne1x+1),1)

fixeddofs -L1: 2*nely+1;

top88(160,100,0.4,3,6,1)

nely/B

lely /2,nelx/3)

F= sparse(L2*(nely+1)+nelx+2, 2*(nely+1)*(nelx+1)J

[12],[1-1],2*(ne1y+1)*(nelx+1),2)

U=zer8(2*(nely+1)*(nelx+1),2);

1y×ne1x

passive zeros (nely, nelx);

U(freedofs, )=K(freedofs, freedofs)\F(freedofs,:

1: nelx

for j =1: nely

if sqrt((i-nely/2)2+(i-nelx/3)"2)< nely/3

c(x)=∑UKU2

end

C=0

dc=o

for i =1: size(F, 2)

Ui=U(:,1);

xPhys(passive==2)=1

ce= reshape(sum((Ui(edofMat)KE).Ui(edofMat), 2)

nely, nelx)

oa Q c+sum(sum((Emin+xPhys. penal*(EO-Emin)).*ce))

88(150,100,0.5,3,5,1)

top88(150,150,0,4,3,6,1)

【实例截图】

【核心代码】